How to Make It
In a saucepan, combine the apples, 1/4 cup of the sugar and 1 cup of the verjus and bring to a boil, stirring, just until the sugar dissolves. Simmer, stirring occasionally, until the apples are softened but not broken down, about 8 minutes. Let the apples cool in the syrup for 15 minutes. Set a strainer over a bowl and drain the apples, reserving the syrup.
Preheat the oven to 350°. Butter and flour a 9-inch springform pan. In a bowl, sift the flour, baking soda, cream of tartar and salt. In a large bowl, using an electric mixer, beat the butter with 1/2 cup of the sugar at medium speed until fluffy. Add the vanilla, then the eggs, 1 at a time; beat well between additions. At low speed, beat in the dry ingredients in 3 batches, alternating with the milk.
Scrape the batter into the prepared pan and smooth the surface. Arrange the apple slices on the batter in concentric circles and press them in halfway. In a bowl, mix 2 teaspoons of the sugar with the cinnamon; sprinkle evenly over the apples. Bake the cake in the middle of the oven for 1 hour, or until golden and risen and a toothpick inserted in the center comes out clean.
Meanwhile, in a medium saucepan, simmer the remaining 1 cup of sugar with the remaining 1 cup of verjus over moderate heat until a deep amber caramel forms, about 12 minutes. Remove from the heat. Immediately whisk in the reserved apple syrup. Brush 1/2 cup of the syrup all over the cake and let cool. Remove the springform ring and transfer the apple cake to a serving plate. Cut into wedges and serve warm, with the remaining syrup and crème fraîche on the side.